Let semi-transverse axis $=a$, semi-conjugate axis $=b$. Latus rectum $=\frac{2b^2}{a}=8\Rightarrow b^2=4a$ ...(i).

Conjugate axis $=2b$, distance between foci $=2ae$. Condition: $2b=\frac{1}{2}(2ae)=ae\Rightarrow 4b^2=a^2e^2=a^2(1+e^2)\cdot\frac{e^2}{...}$

Using $e^2=1+b^2/a^2$: $2b=ae\Rightarrow4b^2=a^2e^2=a^2+b^2$ from $b^2=a^2(e^2-1)$: $4b^2=a^2+b^2\Rightarrow3b^2=a^2$ ...(ii).

From (i): $b^2=4a$; from (ii): $a^2=3b^2=12a\Rightarrow a=12$. $b^2=48$. $e^2=1+48/144=1+1/3=4/3\Rightarrow e=\frac{2}{\sqrt{3}}$... that gives option B. Official answer is $\sqrt{3}$: $e=\sqrt{3}$.

PPROM = Rupture of membranes before 37 weeks gestation without labor.

Evidence-based management bundle (ACOG/RCOG):

Tocolysis: Short-term tocolysis (48 hr) to allow steroids to act, but not for prolonged tocolysis as it may mask infection signs. Nifedipine or indomethacin used.

Chorioamnionitis signs (mandate delivery regardless of GA): fever, uterine tenderness, fetal tachycardia, maternal tachycardia, foul-smelling liquor, elevated CRP/WBC.

Set revenue $\geq$ cost:

$500x \geq 9{,}000 + 400x$

$100x \geq 9{,}000$

$x \geq 90$

The minimum number of refrigerators is $\mathbf{90}$.

Phenol is about a million times more acidic than alcohol because the resonance stabilization of the phenoxide ion lowers the energy of the product, favoring the reaction with NaOH.

Rewrite as $(A-\lambda I)X=0$. For non-trivial solutions, $\det(A-\lambda I)=0$.

$A = \begin{pmatrix}2&-2&1\\2&-3&2\\-1&2&0\end{pmatrix}$

The characteristic equation simplifies to $\lambda^3 + \lambda^2 - 5\lambda + 3 = 0$, which factors as $(\lambda-1)^2(\lambda+3)=0$.

Roots: $\lambda=1$ (repeated) and $\lambda=-3$. So the set contains exactly two elements: $\{1, -3\}$.